im not sure which one goes on what but if the 10 tooth starter seems to start the 16hp kohlers well then i would take that starter and just use the starter gear off of the 16 tooth starter and see if that fixes your problem. if that doesnt work then maybe the 16 tooth gears may bind up if they dont fit correctly so maybe you might want to find a replacement gear for the 10 tooth starter. or instead of wasting time you could take the 2 starter gears and see which starter gear matches your flywheel teeth the best. im not an expert here but thats what i would probably do! hope this helps!

I don't think you could exchange gears between the 2 types. The different diameter of the gears will result in a misalignment when you mount the starter with the different sized gear. You may be able to shim a 10tooth starter to use a 16 tooth gear but it wouldn't work the other way.

Gonna be the 10 tooth. The two shold have diferent diameters, so the 16 won't line up if you try and switch gear heads.

If the starter won't turn with enough authority you could have several issues;

Check the wiring and GROUND to the starter to make sure it's good. Most starters ground to the engine block, so make sure the engine block is grounded good to the frame. I've run a jumper from the negative of the battery to the starter mount bolt before to ENSURE I had a good ground. Wires corrode over time and damage may not be visible....

Starter selinoid may have burned contacts. Not usually, but possible. Use a good set of jumper cables to your battery to test the starter directly - if it works good off the jumper, you have a wiring problem. This works for the first thing as well...

Pull and rebuild your starter. The starter has 2 big magnets inside, glued to the case. One (or both) can be broken and not working. I had a starter that worked but had no power - one of the magnets was fractured and had pieces all over the inside of the starter. Luckily, I had another case laying around that I could use.

Starter has brushes and a commutator - brushes wear out and need to be replaced and the commutator (copper disk on end of the stator) oxidizes and makes poor contact. You can buy replacement brushes and clean the commutator with very fine sand paper (500 grit).

Worst case, the stator has burned/shorted coils in the winding - short (sorry, no pun intended..) of rewinding it, no way to fix that...

After having said all that, replacement starters are around $60 if you shop around - replacement brand, not name brand. An added bonus, most of these replacements have more wire in them and have more torque than the OEM starter. May not last as long as the original, but if you are replacing a 30 year old starter........